As is well understood, the closed loop gain of system may be given by equation 1, in which G(s) is the forward transfer function, H(s) is the feedback transfer function.Out(s)/In(s)=G(s)/(1+HG(s))  Equation 1 As is readily discernable from Equation 1, the closed loop system becomes unstable at any frequency that renders the open loop transfer function to have a unity magnitude (e.g., |HG(s)|=1) and a phase=180°, which results in a divide by zero. Referring to FIG. 1, conventional systems are designed to have a target phase margin to provide suitable stability by avoiding this region. However, variations related to environmental conditions, aging of components, may cause the phase margin provided by a conventional controller to slide up or down the frequency axis leading to phase margin degradation at the target frequency for which it was designed. Thus, a conventional controller that is designed to provide stability can contribute to destabilizing the system.
FIG. 1 shows a graph 100 showing the open loop frequency response of a system showing magnitude versus frequency curves 102a-c on the top and a phase versus frequency curve 104 directly below. The phase margin 105 is defined as the difference between 180° and the phase at the unity gain crossing of the complex open loop transfer function. The gain of the system is subject to fluctuation due to variations in parameters such as temperature, age, and process. The magnitude curve 102a represents the case without parameter variation and magnitude curves 102b and 102c correspond to fluctuations due to variations. These variations alter the unity gain crossing 106a-c, which in turn alter the point on the phase curve 104 that defines that phase margin 105. Thus, the phase margin degrades due to the gain variations, as indicated by the phase margins 105b and 105c as compared to phase margin 105a. 
Thus, conventionally, the open loop gain, HG(s), is designed such that the closed loop system has a target phase and gain margin at the unity gain of the open loop transfer functions. However, variations related to process, temperature, and aging erode the margins and lead to instability. For example, as the phase and gain margins degrade, the system gets closer and closer to the instability region defined by unity gain and phase close to 180°.
In addition to being stable, circuits such as phase lock loops should have other desirable properties such as quick settling times, low overshoot, and fast lock-time. Unfortunately, conventional circuits often are designed with transfer functions that lead to significant overshoot, long settling times, and long lock-times. For example, it can be difficult to design a system that is both stable and has the properties mentioned.